Consider the following list

list={0 < x < 1/12, 0 < x < 1/4}

I wish to find the intersection of the two ranges of x and check if the result is an element of the above list, i.e.

In:=Simplify[And @@ list]

Now I want to check if the above range of x belongs to the list.

in:=MemberQ[list, Simplify[And @@ list]]
out:= False

But It is clear that such a range is an element of list. This is not giving the expected result because the Heads are not matching.

in:=FullForm[Simplify[And @@ list]]


in:=FullForm[0 < x < 1/12]

How to resolve this?

  • $\begingroup$ This is weird. I didn't know it converted to Inequality. A possible workaround is to construct intervals like intervals = Interval[{First[#], Last[#]}] & /@ list; then do result = IntervalIntersection @@ intervals and MemberQ[intervals, result] is true. But this will not work well for arbitrary inequalities, only these two sided ones. $\endgroup$
    – flinty
    Aug 3, 2020 at 16:40
  • 1
    $\begingroup$ Forcing it to expand the inequalities into logical expressions with && seems to work around it too MemberQ[Reduce`InequalityExpand /@ list, Reduce`InequalityExpand[Simplify[And @@ list2]]] $\endgroup$
    – flinty
    Aug 3, 2020 at 17:52
  • $\begingroup$ Yes. After using LogicalExpand, it is working $\endgroup$
    – Antimony51
    Aug 3, 2020 at 18:18

1 Answer 1


Your intent is interval arithmetics. Mathematica offers specialized built-ins to deal with intervals.


IntervalUnion[Interval[{0, 1/12}], Interval[{0, 1/4}]]

Interval[{0, 1/4}]

IntervalIntersection[Interval[{0, 1/12}], Interval[{0, 1/4}]]

Interval[{0, 1/12}]

FullForm@IntervalIntersection[Interval[{0, 1/12}], Interval[{0, 1/4}]]


This is some a tautology and the check corresponding Your MemberQ input is redundant.

On the other hand there is the built-in IntervalMemberQ:

IntervalMemberQ[Interval[{0, 1/12}], Interval[{0, 1/15}]]


To more directly:

Equal[Inequality[0, Less, x, Less, Rational[1, 12]], 
 Less[0, x, Rational[1, 12]]]

(0 < x < 1/12) == (0 < x < 1/12)

The output is not True and


does not alter any.


Equivalent@Equal[Inequality[0, Less, x, Less, Rational[1, 12]], Less[0, x, Rational[1, 12]]]


Equivalent represents the logical equivalence of both representation.

Equivalent[%741] // TautologyQ


TautologyQ@Equal[Inequality[0, Less, x, Less, Rational[1, 12]], 
     Less[0, x, Rational[1, 12]]]


It is in the terms of Mathematica not a direct tautology, but is set with Equal.

NumberLinePlot[{Interval[{0, 1/4}], Interval[{0, 1/12}]}]


This does not work with the Inequality.

NumberLinePlot[Simplify[And @@ list], {x, 0, .1}]


This brings up the real differences between both representation and how there are meant to be made equal in Mathematica.

And it visualizes the main drawback. The interval in Mathematica is closed, the inequalities are strict and therefore open.

The very most suitable answer is use Reduce:

Reduce[list, x]

0 < x < 1/12

Reduce[Inequality[0, Less, x, Less, Rational[1, 12]], x]

0 < x < 1/12

FullForm of both is the very same.

FullForm@Reduce[Inequality[0, Less, x, Less, Rational[1, 12]], x]


FullForm@Reduce[list, x] Inequality[0,Less,x,Less,Rational[1,12]]

FullForm@Reduce[Inequality[0, Less, x, Less, Rational[1, 12]], x]==FullForm@Reduce[list, x]


  • $\begingroup$ Equivalent@Equal[Inequality[0, Less, x, Less, Rational[1, 12]], Less[0, x, Rational[1, 12]]] This isn't doing what you think. Try Equivalent@Equal[1, 0] for example. It returns True. $\endgroup$
    – flinty
    Aug 3, 2020 at 17:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.